On the rank of quadratic twists of elliptic curvers over function fields

2005/03/31 by Emmanuel Kowalski, Kowalski, Emmanuel
Mathematics · #11G05 #11G40 #11R45 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT) #math.NT #msc:11G05 #msc:11G40 #msc:11R45

paper · pdf · doi:10.48550/arxiv.math/0503732

15 pages

arxiv created 2005/03/31 · openalex publication_date 2005/03/31 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove quantitative upper bounds for the number of quadratic twists of a given elliptic curve E/\Fp_q(C) over a function field over a finite field that have rank ≥ 2, and for their average rank. The main tools are constructions and results of Katz and uniform versions of the Chebotarev density theorem for varieties over finite fields. Moreover, we conditionally derive a bound in some cases where the degree of the conductor is unbounded.

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• Ranks of Elliptic Curves Twisted by Quadratic Forms